Option pricing model for event driven call and put options

ABSTRACT

Systems and methods are provided for valuing event driven option contracts. A jump diffusion based model, such as a Merton jump diffusion based model, is modified to assume arithmetic movement of an underlying price and a single jump. The arithmetic movement of the underlying price may be modeled with a Bachelier based arithmetic model. Calculated values may be used to determine margin account requirements.

The present application is a continuation of U.S. patent applicationSer. No. 13/229,237 filed Sep. 9, 2011, which is a continuation-in-partof U.S. patent application Ser. No. 12/245,448 filed Oct. 3, 2008. Theentire disclosure of which is hereby incorporated by reference.

FIELD OF THE INVENTION

The present invention relates to methods and systems for valuing eventdriven option contracts.

DESCRIPTION OF THE RELATED ART

Options contracts or options give their owners the right but not theobligation to buy, in the case of call options, or to sell, in the caseof put options, an underlying good, such as a company's stock or bond,at a specified “strike” price for a preset amount of time. When thepreset amount of time has lapsed, the option “expires.”

Event driven option contracts vary or scale the payment made by theseller or writer of the contract to the buyer depending on how far anevent results from the “strike.” If the event or strike on which anevent driven option contract is based is whether the Board of Governorsof the Federal Reserve will raise an interest rate, the seller only paysthe buyer if the Board of Governors of the Federal Reserve raise theinterest rate, in which case the event driven option contract is in themoney. The payout under the event driven option contract scales and maybe proportionate to the amount of the increase.

Traders, exchanges and other entities determine values for event drivenoption contracts for a number of purposes. Traders often value eventdriven option contracts when making buy and sell decisions. Exchangesand clearing firms value event driven option contracts when determiningmargin requirements. Calculating the value of an event driven optioncontract can be difficult when the option contract has not tradedrecently or frequently.

Prior art approaches to valuing event driven option contracts includedanalytical models and simulation based models that use values ofunderlying financial instruments. The performance of previous analyticalmodels degrades as event driven option contracts become more complex.Simulation models require excessive processing requirements. The valueof an event driven option contract may change as the value of theunderlying product changes. The use of simulation models becomes moreimpractical when the value of the underlying financial instrumentchanges frequently.

Therefore, there is a need in the art for improved analytical systemsand methods for valuing event driven option contracts.

SUMMARY OF THE INVENTION

Embodiments of the present invention overcome problems and limitationsof the prior art by providing systems and methods for valuing eventdriven option contracts that use a jump diffusion based model thatassumes arithmetic movement of an underlying price and a single jump.The jump diffusion model may be based on the Merton jump diffusionmodel. The arithmetic movement of the underlying price may be modeledwith a Bachelier based arithmetic model. In various embodimentscalculated event driven option contract values may be used, for example,when making buy and sell decisions and setting margin requirements.

In other embodiments, the present invention can be partially or whollyimplemented on a computer-readable medium, for example, by storingcomputer-executable instructions or modules, or by utilizingcomputer-readable data structures.

Of course, the methods and systems of the above-referenced embodimentsmay also include other additional elements, steps, computer-executableinstructions, or computer-readable data structures. In this regard,other embodiments are disclosed and claimed herein as well.

The details of these and other embodiments of the present invention areset forth in the accompanying drawings and the description below. Otherfeatures and advantages of the invention will be apparent from thedescription and drawings, and from the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention may take physical form in certain parts and steps,embodiments of which will be described in detail in the followingdescription and illustrated in the accompanying drawings that form apart hereof, wherein:

FIG. 1 shows a computer network system that may be used to implementaspects of the present invention;

FIG. 2 illustrates a price calculation module that may be used tocalculate a value or price of an event driven option contract, inaccordance with an embodiment of the invention; and

FIG. 3 illustrates a method for valuing an event driven option contractin accordance with an embodiment of the invention.

DETAILED DESCRIPTION OF THE INVENTION

Aspects of the present invention may be implemented with computerdevices and computer networks that allow users to perform calculationsand exchange information. An exemplary trading network environment forimplementing trading systems and methods is shown in FIG. 1. An exchangecomputer system 100 receives orders and transmits market data related toorders and trades to users. Exchange computer system 100 may beimplemented with one or more mainframe, desktop or other computers. Auser database 102 includes information identifying traders and otherusers of exchange computer system 100. Data may include user names andpasswords potentially with other information to identify users uniquelyor collectively. An account data module 104 may process accountinformation that may be used during trades. A match engine module 106 isincluded to match bid and offer prices. Match engine module 106 may beimplemented with software that executes one or more algorithms formatching bids and offers. A trade database 108 may be included to storeinformation identifying trades and descriptions of trades. Inparticular, a trade database may store information identifying the timethat a trade took place and the contract price. An order book module 110may be included to compute or otherwise determine current bid and offerprices. A market data module 112 may be included to collect market dataand prepare the data for transmission to users. A risk management module134 may be included to compute and determine a user's risk utilizationin relation to the user's defined risk thresholds. An order processingmodule 136 may be included to decompose variable defined derivativeproduct and aggregate order types for processing by order book module110 and match engine module 106.

The trading network environment shown in FIG. 1 includes computerdevices 114, 116, 118, 120 and 122. Each computer device includes acentral processor that controls the overall operation of the computerand a system bus that connects the central processor to one or moreconventional components, such as a network card or modem. Each computerdevice may also include a variety of interface units and drives forreading and writing data or files. Depending on the type of computerdevice, a user can interact with the computer with a keyboard, pointingdevice, microphone, pen device or other input device.

Computer device 114 is shown directly connected to exchange computersystem 100. Exchange computer system 100 and computer device 114 may beconnected via a T1 line, a common local area network (LAN) or othermechanism for connecting computer devices. Computer device 114 is shownconnected to a radio 132. The user of radio 132 may be a trader orexchange employee. The radio user may transmit orders or otherinformation to a user of computer device 114. The user of computerdevice 114 may then transmit the trade or other information to exchangecomputer system 100.

Computer devices 116 and 118 are coupled to a LAN 124. LAN 124 may haveone or more of the well-known LAN topologies and may use a variety ofdifferent protocols, such as Ethernet. Computers 116 and 118 maycommunicate with each other and other computers and devices connected toLAN 124. Computers and other devices may be connected to LAN 124 viatwisted pair wires, coaxial cable, fiber optics or other media.Alternatively, a wireless personal digital assistant device (PDA) 122may communicate with LAN 124 or the Internet 126 via radio waves. PDA122 may also communicate with exchange computer system 100 via aconventional wireless hub 128. As used herein, a PDA includes mobiletelephones and other wireless devices that communicate with a networkvia radio waves.

FIG. 1 also shows LAN 124 connected to the Internet 126. LAN 124 mayinclude a router to connect LAN 124 to the Internet 126. Computer device120 is shown connected directly to the Internet 126. The connection maybe via a modem, DSL line, satellite dish or any other device forconnecting a computer device to the Internet.

One or more market makers 130 may maintain a market by providing bid andoffer prices for a derivative or security to exchange computer system100. Exchange computer system 100 may also exchange information withother trade engines, such as trade engine 138. One skilled in the artwill appreciate that numerous additional computers and systems may becoupled to exchange computer system 100. Such computers and systems mayinclude clearing, regulatory and fee systems. Coupling can be direct asdescribed or any other method described herein.

The operations of computer devices and systems shown in FIG. 1 may becontrolled by computer-executable instructions stored on acomputer-readable medium. For example, computer device 116 may includecomputer-executable instructions for receiving order information from auser and transmitting that order information to exchange computer system100. In another example, computer device 118 may includecomputer-executable instructions for receiving market data from exchangecomputer system 100 and displaying that information to a user.

Of course, numerous additional servers, computers, handheld devices,personal digital assistants, telephones and other devices may also beconnected to exchange computer system 100. Moreover, one skilled in theart will appreciate that the topology shown in FIG. 1 is merely anexample and that the components shown in FIG. 1 may be connected bynumerous alternative topologies.

FIG. 2 illustrates a price calculation module 202 that may be used tocalculate a value or price of an event driven option contract, inaccordance with an embodiment of the invention. Price calculation module202 may include a memory module 204 and a processor 206. Pricecalculation module 202 may be located at an exchange, such as atexchange computer system 100 (shown in FIG. 1), at a trader workstationremote from an exchange or at any other location that users desire tovalue event driven option contracts. Memory module 204 may beimplemented with one or more physical or magnetic memory devices, suchas a disk drive, magnetic memory, optical disk or other device used tostore computer-executable instructions. In one embodiment, memory module204 is implemented with a random access memory RAM of processor 206.

Memory module 204 includes a model used to value event driven optioncontracts. The model may be a Merton jump diffusion based model 208 thatincludes assumptions 210. Assumptions 210 may include geometric motionof the price of an underlying financial instrument and a finite numberof events, such as one event. An exemplary model is described in detailbelow.

The Merton jump diffusion model generically describes underlying pricemotion as geometric movement dS/S in an underlying instrument S drivenby a composite of the 2 independent processes:

1—time continuous diffusion (Brownian) motion with annualized variance(δ²) cumulative over time to expiration (t) modeled by Wiener processdz, and2—event driven Poisson process dp (converging to Binomial process atlimit) with jump variance (δ²)

dS/S=r*dt+σ*dz+δ*dp,  (equation 1)

-   -   where r is expected return rate, dz=ε*√dt, dρ=1 or 0 (with        probability p or 1−p)    -   ε—normal variable, dt—small increment of t—time to expiration

The total variance over time t, conditioned on discrete number (n) ofjumps is a sum of diffusion variance and event driven jumps (n) variance

s ² t=σ ² t+δ ² *n  (equation 2)

The Merton European call model is a pricing option value (G) with strikeK as a sum of the Black-Scholes (B-S) option values g_(n) weighted withprobabilities (w_(n)) of randomly timed jumps (n) generated in economicevents (m) from a poisson distribution. Underlying price and strikesmust be positive to fit geometric process assumption.

$\begin{matrix}{G = {\sum\limits_{n = 0}^{m->\infty}{w_{n}{g_{n}\left( {S,K,{s(n)},t} \right)}}}} & \left( {{equation}\mspace{14mu} 3} \right)\end{matrix}$

-   -   where w_(n) probabilities (weights) of n jumps occurrence follow        Poisson distribution converging to a binomial at limit when        number of events (m) is large

$\begin{matrix}{{G = {\sum\limits_{n = 0}^{m}{w_{n}g_{n}}}},} & \left( {{equation}\mspace{14mu} 5} \right)\end{matrix}$

-   -   expected number of jumps v=p*m, p is probability of the jump        (jump rate)    -   g₀ Is intrinsic value set as max(S−K, 0) at n=0 (no jumps),    -   g_(n=e) ^(−r*t)(S*N(d)*e^(rt)−K*N(d−s√t)—is a Black-Scholes        value with volatility (s) generated in n>0 jumps    -   and option moneyness d=[ln(S/K)+(r+s²/2)*t]/(s√t), N( ) is        normal cumulative distribution

In accordance with an embodiment of the invention, the Merton jumpdiffusion model is extended to value or price event driven optioncontracts with jumps timed deterministically rather than randomly andunderlying and strike prices limited to a positive range to fitgeometric process assumption. We first reduce an underlying event drivenPoisson process to a binomial one. Then the option value is a weightedsum of intrinsic value and Black-Scholes option values with volatilityrates generated in event driven n>0 jumps

$\begin{matrix}\begin{matrix}{w_{n} = {e^{- v}*{v^{n}/{n!}}}} \\{= {\lim\limits_{m->\infty}\; {{{m!}/{\left( {m - n} \right)!}}*{n!}*p^{n}*\left( {1 - p} \right)^{m - n}}}}\end{matrix} & \left( {{equation}\mspace{14mu} 4} \right)\end{matrix}$

Reducing total variance to events driven variance only by settingdiffusion variance to 0 in event driven auction markets

σ=0  (equation 6)

results in

-   -   g_(n=e) ^(rt)*(S*N(d)−K*N(d−δ √n))—is Black-Scholes value with        volatility (s) generated in n>0 jumps    -   and option moneyness d

d=[ln(S/K)+δ²2*n]/δ √n  (equation 8)

The Merton jump diffusion model is adapted with a Black-Scholes modelwith the number of jumps (n) serving as approximation for the time toexpiration (t). Finally, in the event based auction markets there isgenerally only one deterministically scheduled economic event (m=1) sothat

G=w ₀ g ₀ +w ₁ g ₁, where w ₀=1−p and w ₁ =p  (equation 9)

and the probability rate of a jump (p) is the only tune up parameter.

In accordance with an embodiment of the invention, the Merton jumpdiffusion model is modified with a Bachelier based arithmetic motionmodel. The Merton European call jump diffusion model assumes geometricunderlying process and computes option value as a composite sum ofBlack-Scholes option prices g_(n) with volatility based on bothdiffusion and event driven jumps. The Bachelier model assumes arithmeticmotion in the underlying instrument process driven by diffusion.

dS=σ*dz  (equation 10)

and the Bachelier model computes a European call option as:

a=e− ^(rt)*(s*√t*(d*N(d)+N′(d))),  (equation 11)

-   -   s is volatility    -   and option moneyness d

d=(S−K)/(s √t), N′(d)=n(d)=e ^(−d̂2/2)/√2π—is normal density distribution  (equation 12)

The generic Merton jump diffusion model is modified to price scheduledevent driven option contracts with deterministically timed jumps and aBachelier based approach to underlying arithmetic motion. In presence ofevent driven jumps, underlying arithmetic motion approximation dS hasboth diffusion and jump components and can be described as

dS=σ*dz+δ*dp  (equation 13)

and s²—total variance over time t includes both diffusion variance andevent driven variance generated in n>0 jumps:

s ² t=σ ² t+δ ² *n  (equation 14)

Then the Bachelier value with both diffusion and event driven volatilitygenerated in n>0 jumps is:

a _(n) =e− ^(rt)*(s(n)√n*(d*N(d)+N′(d)))  (equation 15)

Next, the geometric process in the Merton jump diffusion model isreplaced with an arithmetic Bachelier based process. Then valuing anevent driven option contract is a composite sum of Bachelier optionprices a_(n) with volatility based on Event driven n>0 jumps

$\begin{matrix}{A = {\sum\limits_{n = 0}^{m->\infty}{w_{n}a_{n}}}} & \left( {{equation}\mspace{14mu} 16} \right)\end{matrix}$

To model event based option contracts with jumps generated in underlyingeconomic events, we reduce the Poisson process to a binomial process.Then the value is a weighted sum of intrinsic value and option valuesa_(n) with volatility rates generated in event driven n>0 jumps

$\begin{matrix}{A = {\sum\limits_{n = 0}^{m}{w_{n}a_{n}}}} & \left( {{equation}\mspace{14mu} 17} \right)\end{matrix}$

Reducing diffusion variance σ²=0 results in

a _(n) =e− ^(rt)*(δ√n*(d*N(d)+N′(d))) is Bachelier value with volatilitygenerated in n>0 jumps  (equation 18)

-   -   a₀—intrinsic value and option moneyness d

d=(S−K)/(δ√n)

A number of jumps (n) may serve as an approximation for the time toexpiration (t).

Finally, in embodiments that involve the event driven auction marketsthere is only one deterministically scheduled underlying economic event(m=1), so that

A=w ₀ a ₀ +w ₁ a ₁, where w₀=1−p and w ₁ =p  (equation 19)

and probability rate of event jumps (p) is the tune up parameter.

Because of the underlying motion arithmetic assumption, underlying priceand strikes are not limited to a positive range and could be positive,zero or negative as in trade deficit or non-farm payroll statisticsrelated contracts in auction markets.

Traders and other entities often associate “Greek” values with risks.Each Greek estimates the risk for one variable: delta measures thechange in the option price due to a change in the stock price, gammameasures the change in the option delta due to a change in the stockprice, theta measures the change in the option price due to timepassing, vega measures the change in the option price due to volatilitychanging, and rho measures the change in the option price due to achange in interest rates.

Delta, gamma, and vega formulas hold in both geometric (G) andarithmetic (A) pricing cases with t time to expiration being replaced byn—number of jumps and s—annualized volatility replaced by δ—volatilityper jump. So Greeks accounting for event driven jumps may be determinedas follows:

Delta Δ_(G) =e− ^(rt**) N(d) and Δ_(A) =e− ^(rt*) N(d)  (equation 20)

Gamma Γ_(G) =e− ^(rt**) N′(d)/(S*δ√n) and Γ_(A) =e− ^(rt*)N′(d)/(δ√n)  (equation 21)

Vega Y _(G) =e− ^(rt**) N′(d)√n and Y _(A) =e− ^(rt*) N′(d)√n  (equation22)

Rho _(G) =e− ^(rt**) X*n*N(d−δ√n) and Rho_(A) =−a _(n) *n  (equation 23)

Then, similar to options composite Greeks may be produced as weightedsums of Greek values conditioned on jumps generated with jumpprobability rate p.

$\begin{matrix}{{Greek} = {\sum\limits_{n = 0}^{m}{w_{n}{Greek}_{n}}}} & \left( {{equation}\mspace{14mu} 24} \right)\end{matrix}$

And for one deterministically scheduled underlying economic event (m=1)intrinsic Greek₀=0 and so

Greek=w ₁Greek₁, where w ₁ =p  (equation 25)

Greeks jump rate parameter can be tuned up against Black-Scholes modeldetermined Greeks and improve risk analysis when Black-Scholes modeldetermined Greeks (i.e. Gamma) are overestimated at near expirationtime.

Various embodiments of the invention may also utilize a modifiedBachelier model to price (1) put and (2) put and call event basedoptions. For example, the Bachelier model may be extended to priceEuropean style event based call options with δ—Event volatilityreplacing annualized volatility s so that:

c ₁ =e− ^(rt*)((S−K)*(N(d)+N′(d)*δ))  (equation 26)

where d=(S−K)/δ

-   -   S=underlying price    -   K=strike price    -   r=interest rate    -   t=time to expiration

Similarly, a Bachelier model can be adapted to price put event basedoptions

p ₁ e ^(−rt)*((S−K)*(N(d)−1)+N′(d)*δ)  (equation 27)

So either Call or Put option price is given as

c ₁ =e− ^(rt)*(b*(S−K)*(N(b*d)+N′(d)*δ))  (equation 28)

where

-   -   b=1—for a call    -   b=−1—for a put    -   N(x) is Normal Cumulative distribution.

Binary cash-or-nothing options with Q fixed payoff may be modeled asfollows. Note that N(b*d) is probability of option being at or in themoney. Then Binary cash-or-nothing Price is a discounted expected valueof receiving payout Q.

a _(c) =e ^(rt) *Q*N(b*d)  (equation 29)

To derive Greeks recall that derivative of Normal Cumulativedistribution

N′(x)≡n(x)=e ^(−x 2/2)/√2π− is Normal density distribution  (equation30)

And Normal Cumulative and Density distributions partial derivatives are

N _(y)(x)=n(x)*x _(y)  (equation 31)

n _(y)(x)=n(x)*(−x)*x _(y)  (equation 32)

where y=S or δ

Then Binary cash-or-nothing delta is

Δa _(c) =e ^(−rt) *b*Q*n(d)/δ  (equation 33)

Binary cash-or-nothing gamma is

Γa _(c) =e ^(−rt) *b*Q*n _(s)(b*d)/δ=e ^(−rt)*b*Q*n(d)*(−d)/δ²  (equation 34)

Binary cash-or-nothing vega is

Ya _(v) =e ^(−rt) *b*Q*n(d)*(K−S)/δ² =−e ^(−rt) *b*Q*n(d)*d/δ  (equation35)

Binary cash-or-nothing rho is

Ra _(c) =−a _(c) *t  (equation 36)

Binary cash-or-nothing theta is

Ta _(c) =r a _(c) +e ^(−rt) *b*Q*n(d)*0.5*d/t  (equation 37)

where event volatility δ=sS√t.

FIG. 3 illustrates a method for valuing an event driven option contractin accordance with an embodiment of the invention. The method shown inFIG. 3 may be implemented with one or more computer devices. In step 302event driven option contract parameters are received. As shown in FIG.2, event driven option contract parameters may include an underlyingprice, strike price, interest rate, implied volatility, and time toexpiration. In step 304 an event driven option contract is modeled witha jump diffusion based model that assumes arithmetic movement of theprice of an underlying product and a single jump. The modeling of eventdriven option contracts has been described in detail above.

Of course, one or more of the steps shown in FIG. 3 may be performed inan order different than shown. For example, in one embodiment, step 304may be performed before step 302. In step 306, the value of the eventdriven option contract is calculated with the model that results fromstep 304. Once the value of the event driven option contract has beendetermined, a user or computer device may then perform a variety ofadditional steps. For example, in step 308 a margin account requirementmay be determined. Step 308 may involve using the value in a riskscenario calculation such as one used with the Standard PortfolioAnalysis of Risk (SPAN) system, which is used for calculating marginrequirements for futures and options on futures. In other embodiments ofthe invention, the value calculated in step 306 may be displayed on adisplay device and/or used by a user who is making a buy or selldecision.

The present invention has been described herein with reference tospecific exemplary embodiments thereof. It will be apparent to thoseskilled in the art, that a person understanding this invention mayconceive of changes or other embodiments or variations, which utilizethe principles of this invention without departing from the broaderspirit and scope of the invention as set forth in the appended claims.All are considered within the sphere, spirit, and scope of theinvention.

We claim:
 1. A computer-implemented method of valuing an event drivenoption, the method comprising: (a) storing in a memory module a modelfor the event driven option, the model comprising a jump diffusion basedmodel that assumes arithmetic movement of an underlying price and asingle jump; and (b) calculating by a processor the value of the eventdriven option with the model in (a).
 2. The computer-implemented methodof claim 1, wherein the event driven option is based on an interestrate.
 3. The computer-implemented method of claim 2, wherein theinterest rate is set by the Board of Governors of the Federal Reserve.4. The computer-implemented method of claim 1, wherein a payout of theevent driven option is proportionate to an amount of increase in aninterest rate.
 5. The computer-implemented method of claim 1, whereinthe event driven option is based on a non-farm payroll report.
 6. Thecomputer-implemented method of claim 1, wherein (b) comprisesdetermining the value of the event driven option contract fromparameters that include underlying price, strike price, risk freeinterest rate, time to expiration and implied volatility.
 7. Thecomputer-implemented method of claim 1, further including determining amargin account requirement based at least in part on the valuecalculated in (b).
 8. The computer-implemented method of claim 1,further including generating a report with a margin account requirementbase on the value calculated in (b).
 9. The computer-implemented methodof claim 1, wherein the event driven option comprises a European styleoption.
 10. The computer-implemented method of claim 1, wherein the jumpdiffusion model comprises a Merton jump diffusion model.
 11. Anon-transitory computer-readable medium containing computer-executableinstructions for causing a computer device to perform the steps of: (a)modeling an event driven option with a jump diffusion based model thatassumes arithmetic movement and a single jump; and (b) calculating thevalue of the event driven option with the model in (a).
 12. Thenon-transitory computer-readable medium of claim 11, wherein the jumpdiffusion model comprises a Merton jump diffusion model.
 13. Thenon-transitory computer-readable medium of claim 11, wherein a payout ofthe event driven option is proportionate to an amount of increase in aninterest rate.
 14. The non-transitory computer-readable medium of claim13, wherein the interest rate is set by the Board of Governors of theFederal Reserve.
 15. An apparatus that values an event driven option,the apparatus comprising: a computer-readable memory module thatcontains a model of an event driven option, the model comprising a jumpdiffusion based model that assumes arithmetic movement and a singlejump; and a processor configured to receive event based optioncharacteristic data and use the model of the event driven option todetermine a value of the event driven option.
 16. The apparatus of claim15, wherein a payout of the event driven option is proportionate to anamount of increase in an interest rate.
 17. The apparatus of claim 16,wherein the interest rate is set by the Board of Governors of theFederal Reserve.
 18. The apparatus of claim 15, wherein the processor isfurther configured to determine a margin account requirement based atleast in part on the determined value of the event driven option. 19.The apparatus of claim 15, wherein the event driven option is based on anon-farm payroll report.
 20. The apparatus of claim 15, wherein the jumpdiffusion model comprises a Merton jump diffusion model.